Magnetic dipole

A localized magnetic source described at leading nonvanishing dipole order by a magnetic dipole moment. A small current loop is one classical model. The source object, its moment vector and the resulting magnetic dipole field are distinct concepts. At large distance from a source with nonzero dipole moment, higher multipoles give smaller corrections; if the dipole moment vanishes, a higher-order term can lead instead.

A magnetic dipole of fixed magnitude $m_B$, free to rotate about an axis with moment of inertia $J$, has potential energy $-m_{B}B\cos \theta$ in a uniform magnetic field. Near stable alignment, $J\ddot{\theta }=-m_{B}B\theta$, so its small-angle period is $2\pi \sqrt{J/(m_{B}B)}$. This neglects damping, bearing friction and any other restoring torque.

For a light nonmagnetic rod of length $l$ wound uniformly with $N$ thin turns of mean radius $a$, the magnetic dipole moment is $m_B=NI\pi a^2$. Wire of cross-sectional area $A_w$ and density $\rho$ has mass approximately $M=2\pi aN\rho A_w$. The transverse central moment of inertia is $J=M(l^2/12+a^2/2)$ in the thin-winding approximation. Consequently the oscillation period in a transverse magnetic field does not depend on the number of turns: wire inertia and magnetic moment both scale with $N$.

A generic instrument for force and torque uses a magnetically responsive bead in a controlled magnetic field, with optical tracking. Translational force requires a field gradient. Permanent or anisotropic moments can feel orienting torque; an isotropic induced moment parallel to the field has no such torque. This definition concerns the physical instrument, without biological applications.

For a tether with extension $\ell$ and tensile load $F$, a small transverse coordinate costs $F{x}^2/(2\ell )$. Equipartition gives $F=k_{B}T\ell /⟨x^2⟩$ for one Cartesian transverse coordinate. Using the sum of two transverse variances changes the factor of two. Geometry and measurement filtering must match the calibration model.

For an equilibrium overdamped harmonic coordinate with covariance $⟨x(t)x(0)⟩=V{e}^{-|t|/\tau }$, a camera exposure $\Delta$ records the average over that interval. Integrating the covariance twice gives $\mathrm{Var}(\bar{x})=2V{\Delta }^{-2}{\int }_0^{\Delta }(\Delta -u)e^{-u/\tau }du$. The displayed reduction follows with $s=\Delta /\tau$. In the tether calibration $F=k_{B}Th/V$, ignoring exposure averaging overestimates the inferred force. Independent localization-noise variance instead adds to the measured variance and biases it in the opposite direction. Other filters require their actual response functions.

For a linear isotropic unsaturated effective susceptibility contrast $\Delta \chi$, the induced-moment energy is $-V\Delta \chi B^2/(2{\mu }_0)$, giving $\mathbf{F}=V\Delta \chi \nabla B^2/(2{\mu }_0)$. The one-half differs from the fixed permanent-dipole energy because the applied field creates the moment. Demagnetization and saturation are absorbed into or replace the assumed response model.